\Chapter{Results and Discussion}
\label{chap:results}
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\section{Numerical simulation of a plasmonic antenna}
\label{sec:Simulations}

I performed some preliminary investigations into the required parameters for simulating a nanometer-scale plasmonic \BT\ antenna, and subsequently performed a series of simulations to verify experimental results\footnote{experimental work carried out by Yury Alaverdyan and Nick Vamivakas} for the modification of the antenna spectrum upon introduction of a dielectric nanocrystal. 

As previously noted in \fref{sec:FDTD}, the grid spacing $\Delta x$ used in \FDTD\ simulations must be small enough to resolve both the wavelength of interest and the smallest spatial variations in the structure. However, using a smaller cell results in higher memory requirements and longer computation times, so it is desirable to keep the mesh cell size as small as possible while still obtaining representative results. Since the FDTD method is an exact solution of the macroscopic Maxwell equations in the limit $\Delta x\rightarrow 0$, an appropriate cell size for a given structure class can be determined by varying it, and testing for convergence of the quantity of interest, such as spectral response or angular scattering pattern. I performed such a convergence test on a \BT\ antenna structure, prior to running a series of simulations of variations on this structure. The results of the simulations are displayed in \fref{fig:FDTDmeshsizes}. The mesh cell size is fixed over the \BT\ region, but outside the structure, the mesh can be safely relaxed to larger cell sizes to ease computation requirements. The peak wavelength of the spectral response fluctuates around the final solution as the mesh cell size is decreased, but ultimately converges. The memory requirements and processing time rise rapidly for reducing mesh cell size, as expected (since the number of cells is $\propto(\Delta x)^{-3}$). A reasonable compromise between accuracy of results and simulation requirements appeared to be $\Delta x=0.5\unitspace{\rm nm}$. However, I also used a \nm{1} mesh cell for faster preliminary runs to determine appropriate frequency ranges over which to record data. A \nm{0.5} mesh cell size limits the largest structure we can simulate in a reasonable time to a volume of the order of $250\times250\times100\unitspace{\rm nm}$.

\begin{figure}%
\begin{center}
	\leavevmode
	\includegraphics[width=0.9\textwidth]{FDTDmeshsizes}%
	\caption{Results of FDTD simulations for varying mesh cell sizes. Part A  displays the rapidly increasing time and memory requirements of the simulations. Part B displays the peak scattering wavelength, showing convergence to and oscillation around a limiting value.}%
	\label{fig:FDTDmeshsizes}%
\end{center}
\end{figure}

\begin{figure}%
\begin{center}
	\leavevmode
	\includegraphics[width=0.9\textwidth]{BT}%
	\caption{A Bowtie antenna. Part A shows an SEM image of a typical antenna of the type I simulated. Part B shows the CAD rendering of the model I used for my FDTD simulations. The corners are rounded to a radius of \nm{9} in order to more closely approximate the real structure.}%
	\label{fig:BT}%
\end{center}
\end{figure}

Additionally, as also mentioned in \fref{sec:FDTD}, dimensions approaching the size at which quantum mechanical effects become significant cannot be modelled successfully. For the majority of cases I have looked at, this is not a significant issue. However, I did discover that the \nm{1-5} Cr adhesion layer placed below the thermally deposited gold (see \fref{sec:EBL}) must be omitted from the simulation in order to avoid a broadening of the simulated scattering spectra to around twice the line-width observed experimentally.

The aim of this work was to show the modification of an antennas resonance caused by placing a dielectric object in the vicinity of the antenna. The bowtie (BT) antenna design consists of two gold triangles with opposing tips, with a small gap between them. The structures used in this study (visible in \fref{fig:BT}), consist of \nm{75} long triangles, with a gap of \nm{25}. The \BT\ resonance is structure-dependent, this design being chosen to obtain a peak wavelength at around \nm{670}. The scattering spectra from multiple antennae show an uncertainty of the central wavelength of approximately \nm{20}, due to irregularities in fabrication. The \BT\ antenna is a widely used plasmonic antenna design notable for the highly enhanced field confined to the region of the gap. Fields here can reach up to $10^3$ or more times the strength away from the antenna, and are confined to a region only \nm{20} or so in diameter, much smaller than the free-space diffraction limit at this optical wavelength.

\begin{figure}%
\begin{center}
\leavevmode
\includegraphics[width=0.9\textwidth]{BTDNCprofiles}%
\caption{Atomic force microscope (AFM) images of the bowtie-nanocrystal complex, along with FDTD-simulated near-field profiles. Parts A-E show AFM images of the nanocrystal in its different locations. lateral resolution exhibits a \nm{\sim 10} uncertainty due to the convolution of the tip shape with the sample. Parts F-I show near-field profiles of the electric field intensity at substrate level for four of the locations. Intensities are plotted on a logarithmic colour scale, displayed to the right of the figure. Parts G-M show near-field intensity profiles of the same locations, at \nm{17.5} above the substrate surface, where the crystal is at its greatest extent. They use the same logarithmic colour scale as parts F-I. The crystal in images F-M is outlined in white to aid the eye. The near-field profiles clearly show the distortion of the local field profile caused by the addition of the nanocrystal, whose high refractive index causes a significant modification to the antenna's local dielectric environment.}%
\label{fig:BTDNCprofiles}%
\end{center}
\end{figure}

A diamond nanocrystal of \nm{\sim\!35} diameter was positioned close to the antenna at a variety of distances using an \AFM, and both dark-field scattering and photoluminescence spectra recorded for each crystal position. \AFM\ images of the different positions are displayed in \fref{fig:BTDNCprofiles}, along with profiles of the electric field strength obtained from my FDTD simulations, on a logarithmic colour scale. Spectral results in \fref{fig:BTDNCspectra} show a significant shift of the resonant wavelength is observed when the diamond nanocrystal (DNC) is positioned close to the gap. FDTD near-field images reveal that the introduction of the crystal in this position also causes the local field profile to be altered, drawn into the high refractive index medium. The FDTD simulations show reasonable agreement with the experimentally measured spectra.

\begin{figure}%
\begin{center}
\leavevmode
\includegraphics[width=120mm]{btdncSpectra}%
\caption{Spectra of the bowtie-nanocrystal complex shown in \fref{fig:BTDNCprofiles}. The three sets of curves are for FDTD simulations, darkfield scattering and photoluminescence, respectively. }%
\label{fig:BTDNCspectra}%
\end{center}
\end{figure}


%\begin{figure}%
%\begin{center}
%\leavevmode
%\includegraphics[width=0.9\textwidth]{nearfield}%
%\caption{FDTD-simulated near-field profiles of the electric field intensity in a BT antenna alone (a) and with an adjacent diamond nanocrystal (b). The profiles were recorded at frequencies of \unit{448.1}{THz} and \unit{430.7}{THz} respectively, corresponding to vacuum wavelengths of \nm{669} and \nm{696}. The profiles are plotted on a logarithmic colour scale, displayed at the top of the figure. }%
%\label{fig:BTDNCpositions}%
%\end{center}
%\end{figure}

%In this section I will discuss:
%\begin{itemize}
%	\item \done{time \& resource requirement dependence on mesh size}
%	\item \done{spectral convergence with decreasing mesh size}
%	\item numerical simulations from the BTDNC paper including spectra \& near-field profiles
%	\item future simulations I may perform
%\end{itemize}


\section{Waveguide characterization}
\label{Waveguides}
As a preliminary to producing smooth waveguides, and in order to compare the \EBL\ and \FIB\ fabrication techniques, measurements were made of the transmission of light through rough gold waveguides of varying length and widths. Our waveguides consisted of \nm{50} thick Au on top of a \nm{5} Cr adhesion layer, on a $128\circ$ Y-cut lithium Niobate substrate. Lithium Niobate has a high piezoelectric coupling coefficient for the generation of surface acoustic waves, and so is a likely candidate for substrates for later active switching experiments. Since the dielectric environment of plasmonic structures has a significant effect on their mode structures, it was deemed best to use the same substrates for all related tests. Waveguides with lengths from \um{4-20} and widths of \um{2-10} were produced in large arrays on a single substrate using \EBL.

We chose to perform our experiment using a \nm{780} diode laser for excitation. The confocal microscope setup similar to that detailed in \fref{fig:confocal} was used, with the additional modification that the excitation and collection points were displaced relative to each other, such that when excitation was at one end of the waveguide, the collection would be from a point at the opposite end (see inset). No special structures were included to couple excitation beam into the waveguide plasmon mode, or couple the mode out into free space again, instead relying on edge-coupling effects combined with the wide range of $k$ vectors created by the high numerical aperture (NA 0.9) objective used. Although the absolute coupling efficiency for this configuration is unlikely to be very high, since we only desire to measure the decay length, it is a much simpler setup, and is sufficient for our purposes. Recalling from \fref{sec:analytic} that the surface plasmon polariton mode for a planar interface is a purely \TM\ wave, we can expect the excitation of the plasmon mode of the waveguide to be dependent on the polarization of the exciting beam. Although theoretical calculations of the modes of finite-width waveguides bounded by asymmetric dielectrics indicate that none of the modes are purely \TM\ polarized, they suggest that for waveguide with a high aspect ratio (i.e. where width $w\gg t$), the modes are dominated by a \TM\ component.

Parts A and B of \fref{fig:wgScans} show two scans of a single waveguide, with excitation polarization parallel and perpendicular to the waveguide axis, respectively, corresponding to \TM\ and \TE\ polarizations. A clear vertical stripe is visible in part A, for \TM\ polarization, indicating coupling into the plasmon mode from one waveguide end, and collection from the other. In part B, it is clear that excitation in \TE\ polarization couples much less efficiently to the waveguide modes, confirming the interpretation of the result as plasmon mode coupling. Part C of \fref{fig:wgScans} shows a scan in which the separation of the excitation and collection spots is less than the waveguide length. Although this scan shows the same bright feature for excitation from the edge, it also shows coupling from excitation of the waveguide surface. This coupling is a result of the surface roughness of the film, which can act in a similar manner to the edge, providing sufficient momentum to couple the excitation light into the waveguide mode. %In addition, some additional coupling is seen when the excitation scatters of the far edge of the waveguide, with collection from the substrate surface. This is probably a result of direct radiative coupling.

\begin{figure}%
\begin{center}
\leavevmode
\includegraphics[width=\textwidth]{wgScans}%
\caption{Three scans of waveguides, under different conditions. The leftmost scan was performed with excitation and collection separated by the waveguide length, and polarization along the waveguide axis. It shows a bright line where light is coupled into the waveguide at one end and out at the other. Some background scattering is also visible when the excitation is over the waveguide edge while the collection is on the substrate, but this is minimal. The centre scan was performed on the same waveguide, with a smaller separation between excitation and collection, and polarization again along the waveguide axis. The bright smear shows that excitation couples into the waveguide even away from the edge, presumably through surface roughness effects. The rightmost scan shows a scan of a waveguide performed with excitation and collection separated by the waveguide length, but with polarization perpendicular to the waveguide axis, as shown underneath. The image contrast is poor, with no discernible transmission peak, indicating that this polarization does not couple effectively into the waveguide mode.}%
\label{fig:wgScans}%
\end{center}
\end{figure}

\Fref{fig:wgPropLength} shows the log of collected intensity, normalized for input power, from waveguides of a variety of lengths. The straight line fit shows a clear exponential dependence of transmitted intensity, as expected for plasmonic waveguides. The fit gives an attenuation length (the length over which the propagating intensity falls by a factor $e^{-1}$) of \um{\sim 5}. This is in agreement with the attenuation lengths for stripe waveguides measured by others \TODO{references to constants measured by others... also bear inmind ours are on LiNbO$_3$}. The coupling seen in part B of \fref{fig:wgScans} for excitation away from the waveguide edge indicates that coupling into the plasmon mode occurs due to surface roughness even away from the waveguide edge. This also implies that the waveguide mode  has losses increased by its roughness. The elimination of this roughness is the next step in producing more efficient waveguide structures.

\begin{figure}%
\begin{center}
\leavevmode
\includegraphics[width=0.8\textwidth]{wgPropLength6}%
\caption{Plot of the natural log of the transmission efficiency versus waveguide length for two different widths. The blue crosses show data points for \um{4} wide waveguides, the green crosses data points for \um{6} wide waveguides. The linear progressions indicate the expected exponential decay of transmitted intensity with propagation distance. The fits indicate an attenuation length of \um{\sim\!5} for the \um{4} waveguides, and \um{\sim\!7} for the \um{6} guides. the \um{4um} long point was not considered for the fit, as it cannot be properly considered a waveguide in the same way as the other stipes, since its width is equal to its length.}%
\label{fig:wgPropLength}%
\end{center}
\end{figure}



\section{New structures}
\label{NewStructures}
Photoluminescence (PL) measurements provide a simple way of determining whether or not a gold film is rough. Due to the 1:1 mapping of $k$ vectors to energy values in bulk monocrystalline gold, \PL\ ordinarily requires a specific change in momentum from the exciting photon to the emitted photon. Since the crystal boundaries and surface roughness create a local relaxation of the band structure of the metal, they allow PL to occur much more easily. For this reason, rough gold films illuminated by a green \nm{532} laser exhibit much stronger emission at longer wavelengths than smooth films. Although not sufficient for a quantitative analysis of surface roughness, a suppression of PL relative to thermally deposited films is expected if the gold flakes we have produced are truly smooth. Our microscope (see \fref{fig:confocal}) was set up for PL measurements by adding a \nm{532} laser-line filter into the excitation arm, and a \nm{600} long-pass filter into the collection arm. The results of two scans of individual flakes can be seen in parts C and D of \fref{fig:flakes}. The flake is barely visible at all, with only a slight increase in PL collected from the flake edges compared to the surrounding substrate. These initial results, while not conclusive, point to strongly reduced fluorescence from chemically grown gold flakes. This would be beneficial for antenna structures cut from flakes, and indicates potentially lower radiative losses from roughness, which could allow longer propagation lengths for waveguides fabricated from such flakes.

\begin{figure}%
\begin{center}
\leavevmode
\includegraphics[width=\textwidth]{flakes}%
\caption{Chemically grown Au flakes. Part A shows a PL scan of a thermally-deposited gold marker. The image contrast is good, showing strong fluorescence. Part B shows an SEM image of micron-scale monocrystalline gold flakes, chemically synthesized by Yury Alaverdyan. Parts C and D show PL scans I took of flakes similar to those in part B, although larger (the flakes in parts C and D are around \um{10} across). The flakes are triangular, with truncated corners, and are barely discernible in PL, so have been surrounded in red to aid the eye. The twio bright spots in the scan of part D are smaller particles which have stuck to the larger flake. They may be colloidal gold, or a contaminent introduced after production. The suppressed PL, with only a faint suggestion of the edges visible, indicates strongly reduced fluorescence, promising for new structures.}%
\label{fig:flakes}%
\end{center}
\end{figure}

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